Book description
Engineering Mathematics covers the four mathematics papers that are offered to undergraduate students of engineering. With an emphasis on problemsolving techniques and engineering applications, as well as detailed explanations of the mathematical concepts, this book will give the students a complete grasp of the mathematical skills that are needed by engineers.
Table of contents
 Cover
 Title Page
 Contents
 Dedication
 Preface
 Symbols and Basic Formulae

Part I

1. Sequences and Series
 1.1 Sequences
 1.2 Convergence of Sequences
 1.3 The Upper and Lower Limits of a Sequence
 1.4 Cauchy's Principle of Convergence
 1.5 Monotonic Sequence
 1.6 Theorems on Limits
 1.7 Subsequence
 1.8 Series
 1.9 Comparison Tests
 1.10 D’ Alembert's Ratio Test
 1.11 Cauchy's Root Test
 1.12 Raabe's Test
 1.13 Logarithmic Test
 1.14 De Morgan – Bertrand Test
 1.15 Gauss's Test
 1.16 Cauchy's Integral Test
 1.17 Cauchy's Condensation Test
 1.18 Kummer's Test
 1.19 Alternating Series
 1.20 Absolute Convergence of a Series
 1.21 Convergence of the Series of the Type
 1.22 Derangement of Series
 1.23 Nature of Nonabsolutely Convergent Series
 1.24 Effect of Derangement of Nonabsolutely Convergent Series
 1.25 Uniform Convergence
 1.26 Uniform Convergence of a Series of Functions
 1.27 Properties of Uniformly Convergent Series
 Exercises
 2. Mean Value Theorems and Expansion of Functions

3. Curvature
 3.1 Radius of Curvature of Intrinsic Curves
 3.2 Radius of Curvature for Cartesian Curves
 3.3 Radius of Curvature for Parametric Curves
 3.4 Radius of Curvature for Pedal Curves
 3.5 Radius of Curvature for Polar Curves
 3.6 Radius of Curvature at the Origin
 3.7 Centre of Curvature
 3.8 Evolutes and Involutes
 3.9 Equation of the Circle of Curvature
 3.10 Chords of Curvature Parallel to the Coordinate Axes
 3.11 Chord of Curvature in Polar Coordinates
 Exercises

4. Asymptotes and Curve Tracing
 4.1 Determination of Asymptotes When the Equation of the Curve in Cartesian form is given
 4.2 The Asymptotes of the General Rational Algebraic Curve
 4.3 Asymptotes parallel to the Coordinate Axes
 4.4 Working Rule for Finding Asymptotes of Rational Algebraic Curve
 4.5 Intersection of a Curve and its Asymptotes
 4.6 Asymptotes by Expansion
 4.7 Asymptotes of the Polar Curves
 4.8 Circular Asymptotes
 4.9 Curve Tracing (Cartesian Equations)
 4.10 Curve Tracing (Polar Equations)
 4.11 Curve Tracing (Parametric Equations)
 Exercises

5. Partial Differentiation
 5.1 Continuity of a Function of Two Variables
 5.2 Differentiability of a Function of Two Variables
 5.3 The Differential Coefficients
 5.4 Distinction Between Derivatives and Differential Coefficients
 5.5 HigherOrder Partial Derivatives
 5.6 Envelopes and Evolutes
 5.7 Homogeneous Functions and Euler's Theorem
 5.8 Differentiation of Composite Functions
 5.9 Transformation From Cartesian to Polar Coordinates and Vice Versa
 5.10 Taylor's Theorem For Functions of Several Variables
 5.11 Extreme Values
 5.12 Lagrange's Method of Undetermined Multipliers
 5.13 Jacobians
 5.14 Properties of Jacobian
 5.15 Necessary and Sufficient Conditions for Jacobian to Vanish
 5.16 Differentiation Under the Integral Sign
 Exercises
 6. Beta and Gamma Functions

7. Reduction Formulas
 7.1 Reduction Formulas for ∫sinn xdx and ∫cosn xdx
 7.2 Reduction Formulas for ∫sinm x cosn x dx
 7.3 Reduction Formulas for ∫tann xdx and ∫secn xdx
 7.4 Reduction Formulas for ∫xn and ∫xn cos mxdx
 7.5 Reduction Formulas for ∫ xn eax dx and ∫ xm (log x)n dx
 7.6 Reduction Formula for ∫cosm x sin nxdx
 Exercises

8. Volumes and Surfaces of Solids of Revolution
 8.1 Volume of the solid of Revolution (Cartesian Equations)
 8.2 Volume of the Solid of Revolution (Parametric Equations)
 8.3 Volume of the Solid of Revolution (Polar Curves)
 8.4 Surface of the Solid of Revolution (Cartesian Equations)
 8.5 Surface of the Solid of Revolution (Parametric Equations)
 8.6 Surface of the Solid of Revolution (Polar Curves)
 Exercises

9. Multiple Integrals
 9.1 Double Integrals
 9.2 Properties of a Double Integral
 9.3 Evaluation of Double Integrals (Cartesian Coordinates)
 9.4 Evaluation of Double Integrals (Polar Coordinates)
 9.5 Change of Variables in a Double Integral
 9.6 Change of Order of Integration
 9.7 Area Enclosed by Plane Curves (Cartesian and Polar Coordinates)
 9.8 Volume and Surface Area as Double Integrals
 9.9 Triple Integrals and their Evaluation
 9.10 Change to Spherical Polar Coordinates from Cartesian Coordinates in a Triple Integral
 9.11 Volume as a Triple Integral
 Exercises

10. Vector Calculus
 10.1 Differentiation of a Vector
 10.2 Partial Derivatives of a Vector Function
 10.3 Gradient of a Scalar Field
 10.4 Geometrical Interpretation of a Gradient
 10.5 Properties of a Gradient
 10.6 Directional Derivatives
 10.7 Divergence of a VectorPoint Function
 10.8 Physical Interpretation of Divergence
 10.9 Curl of a VectorPoint Function
 10.10 Physical Interpretation of Curl
 10.11 The Laplacian Operator ∇2
 10.12 Properties of Divergence and Curl
 10.13 Integration of Vector Functions
 10.14 Line Integral
 10.15 Work Done by a Force
 10.16 Surface Integral
 10.17 Volume Integral
 10.18 Gauss's Divergence Theorem
 10.19 Green's Theorem in a Plane
 10.20 Stoke's Theorem
 Exercises

11. ThreeDimensional Geometry
 11.1 Coordinate Planes
 11.2 Distance Between Two Points
 11.3 Direction Ratios and Direction Cosines of a Line
 11.4 Section Formulae—Internal division of a line by a point on the line
 11.5 Straight Line in Three Dimensions
 11.6 Angle Between Two Lines
 11.7 Shortest Distance Between Two Skew Lines
 11.8 Equation of a Plane
 11.9 Equation of a Plane Passing Through a Given Point and Perpendicular to a Given Direction
 11.10 Equation of a Plane Passing Through Three Points
 11.11 Equation of a Plane Passing Through a Point and Parallel to Two Given Vectors
 11.12 Equation of a Plane Passing Through Two Points and Parallel to a Line
 11.13 Angle Between Two Planes
 11.14 Angle Between a Line and a Plane
 11.15 Perpendicular Distance of a Point From a Plane
 11.16 Planes Bisecting the Angles Between Two Planes
 11.17 Intersection of Planes
 11.18 Planes Passing Through the Intersection of Two Given Planes
 11.19 Sphere
 11.20 Equation of a Sphere Whose Diameter is the Line Joining Two Given Points
 11.21 Equation of a Sphere Passing Through Four Points
 11.22 Equation of the Tangent Plane to a Sphere
 11.23 Condition of Tangency
 11.24 Angle of Intersection of Two Spheres
 11.25 Condition of Orthogonality of Two Spheres
 11.26 Cylinder
 11.27 Equation of a Cylinder with given Axis and Guiding Curves
 11.28 Right Circular Cylinder
 11.29 Cone
 11.30 Equation of a Cone with Vertex at the Origin
 11.31 Equation of a Cone with Given Vertex and Guiding Curve
 11.32 Right Circular Cone
 11.33 Right Circular Cone with Vertex (α, β, γ), Semi Vertical Angle θ, and (l, m, n) the Direction Cosines of the Axis
 11.34 Conicoids
 11.35 Shape of an Ellipsoid
 11.36 Shape of the Hyperboloid of One Sheet
 11.37 Shape of the Hyperboloid of Two Sheets
 11.38 Shape of the Elliptic Cone
 11.39 Intersection of a Conicoid and a Line
 11.40 Tangent Plane at a Point of Central Conicoid
 11.41 Condition of Tangency
 11.42 Equation of Normal to the Central Conicoid at Any Point (α, β, γ) On It
 Exercises

1. Sequences and Series

Part II

12. Preliminaries
 12.1 Sets and Functions
 12.2 Continuous and Piecewise Continuous Functions
 12.3 Derivability of a Function and Piecewise Smooth Functions
 12.4 The Riemann Integral
 12.5 The Causal and Null Functions
 12.6 Functions of Exponential Order
 12.7 Periodic Functions
 12.8 Even and Odd Functions
 12.9 Sequence and Series
 12.10 Series of Functions
 12.11 Partial Fraction Expansion of a Rational Function
 12.12 Special Functions
 12.13 The Integral Transforms

13. Linear Algebra
 13.1 Concepts of Group, Ring, and Field
 13.2 Vector Space
 13.3 Linear Transformation
 13.4 Linear Algebra
 13.5 Rank and Nullity of a Linear Transformation
 13.6 Matrix of a Linear Transformation
 13.7 Normed Linear Space
 13.8 Inner Product Space
 13.9 Matrices
 13.10 Algebra of Matrices
 13.11 Multiplication of Matrices
 13.12 Associtative Law for Matrix Multiplication
 13.13 Distributive Law for Matrix Multiplication
 13.14 Transpose of a Matrix
 13.15 Symmetric, Skewsymmetric, and Hermitian Matrices
 13.16 Lower and Upper Triangular Matrices
 13.17 Adjoint of a Matrix
 13.18 The Inverse of a Matrix
 13.19 Methods of Computing Inverse of a Matrix
 13.20 Rank of a Matrix
 13.21 Elementary Matrices
 13.22 Equivalence of Matrices
 13.23 Row and Column Equivalence of Matrices
 13.24 Row Rank and Column Rank of a Matrix
 13.25 Solution of System of Linear Equations
 13.26 Solution of Nonhomogeneous Linear System of Equations
 13.27 Consistency Theorem
 13.28 Homogeneous Linear Equations
 13.29 Characteristic Roots and Vectors
 13.30 The CayleyHamilton Theorem
 13.31 Algebraic and Geometric Multiplicity of an Eigenvalue
 13.32 Minimal Polynomial of a Matrix
 13.33 Orthogonal, Normal, and Unitary Matrices
 13.34 Similarity of Matrices
 13.35 Triangularization of an Arbitrary Matrix
 13.36 Quadratic Forms
 13.37 Diagonalization of Quadratic Forms
 Exercises
 14. Functions of Complex Variables

15. Differential Equations
 15.1 Definitions and Examples
 15.2 Formulation of Differential Equation
 15.3 Solution of Differential Equation
 15.4 Differential Equations of First order
 15.5 Separable Equations
 15.6 Homogeneous Equations
 15.7 Equations Reducible to Homogeneous Form
 15.8 Linear Differential Equations
 15.9 Equations Reducible to Linear Differential Equations
 15.10 Exact Differential Equation
 15.11 The Solution of Exact Differential Equation
 15.12 Equations Reducible to Exact Equation
 15.13 Applications of First Order and First Degree Equations
 15.14 Linear Differential Equations
 15.15 Solution of Homogeneous Linear Differential Equation with Constant Coefficients
 15.16 Complete Solution of Linear Differential Equation with Constant Coefficients
 15.17 Method of Variation of Parameters to Find Particular Integral
 15.18 Differential Equations with Variable Coefficients
 15.19 Simultaneous Linear Differential Equations with Constant Coefficients
 15.20 Applications of Linear Differential Equations
 15.21 MassSpring System
 15.22 Simple Pendulum
 15.23 Solution in Series
 15.24 Bessel's Equation and Bessel's Function
 15.25 Legendre's Equation and Legendre's Polynomial
 15.26 Fourier–Legendre Expansion of a Function
 Exercises

16. Partial Differential Equations
 16.1 Formulation of Partial Differential Equation
 16.2 Solutions of a Partial Differential Equation
 16.3 Nonlinear Partial Differential Equations of the First Order
 16.4 Charpit's Method
 16.5 Some Standard forms of Nonlinear Equations
 16.6 The Method of Separation of Variables
 16.7 OneDimensional Heat Equation
 16.8 OneDimensionalWave Equation
 16.9 TwoDimensional Heat Equation
 Exercises

17. Fourier Series
 17.1 Trigonometric Series
 17.2 Fourier (or Euler) Formulae
 17.3 Periodic Extension of a Function
 17.4 Fourier Cosine and Sine Series
 17.5 Complex Fourier Series
 17.6 Spectrum of Periodic Functions
 17.7 Properties of Fourier Coefficients
 17.8 Dirichlet's Kernel
 17.9 Integral Expression for Partial Sums of a Fourier Series
 17.10 Fundamental Theorem (Convergence Theorem) of Fourier Series
 17.11 Applications of Fundamental Theorem of Fourier Series
 17.12 Convolution Theorem for Fourier Series
 17.13 Integration of Fourier Series
 17.14 Differentiation of Fourier Series
 17.15 Examples of Expansions of Functions in Fourier Series
 17.16 Signals and Systems
 17.17 Classification of Signals
 17.18 Classification of Systems
 17.19 Response of a Stable Linear Time Invariant Continuous Time System (LTC System) to a Piecewise Smooth and Periodic Input
 17.20 Application to Differential Equations
 17.21 Application to Partial Differential Equations
 Exercises

18. Fourier Transform
 18.1 Fourier Integral Theorem
 18.2 Fourier Transforms
 18.3 Fourier Cosine and Sine Transforms
 18.4 Properties of Fourier Transforms
 18.5 Solved Examples
 18.6 Complex Fourier Transforms
 18.7 Convolution Theorem
 18.8 Parseval's Identities
 18.9 Fourier Integral Representation of a Function
 18.10 Finite Fourier Transforms
 18.11 Applications of Fourier Transforms
 18.12 Application to Differential Equations
 18.13 Application to Partial Differential Equations
 Exercises

19. Discrete Fourier Transform
 19.1 Approximation of Fourier Coefficients of a Periodic Function
 19.2 Definition and Examples of DFT
 19.3 Inverse DFT
 19.4 Properties of DFT
 19.5 Cyclical Convolution and Convolution Theorem for DFT
 19.6 Parseval's Theorem for the DFT
 19.7 Matrix form of the DFT
 19.8 NPoint Inverse DFT
 19.9 Fast Fourier Transform (FFT)
 Exercises
 20. Laplace Transform

21. Inverse Laplace Transform
 21.1 Definition and Examples of Inverse Laplace Transform
 21.2 Properties of Inverse Laplace Transform
 21.3 Partial Fractions Method to Find Inverse Laplace Transform
 21.4 Heaviside's Expansion Theorem
 21.5 Series Method to Determine Inverse Laplace Transform
 21.6 Convolution Theorem
 21.7 Complex Inversion Formula
 Exercises
 22. Applications of Laplace Transform

23. The ztransform
 23.1 Some Elementary Concepts
 23.2 Definition of ztransform
 23.3 Convergence of ztransform
 23.4 Examples of ztransform
 23.5 Properties of the ztransform
 23.6 Inverse ztransform
 23.7 Convolution Theorem
 23.8 The Transfer Function (or System Function)
 23.9 Systems Described by Difference Equations
 Exercises

24. Elements of Statistics and Probability
 24.1 Measures of Central Tendency
 24.2 Measures of Variability (Dispersion)
 24.3 Measure of Skewness
 24.4 Measures of Kurtosis
 24.5 Covariance
 24.6 Correlation and Coefficient of Correlation
 24.7 Regression
 24.8 Angle Between the Regression Lines
 24.9 Probability
 24.10 Conditional Probability
 24.11 Independent Events
 24.12 Probability Distribution
 24.13 Mean and Variance of a Random Variable
 24.14 Binomial Distribution
 24.15 Pearson's Constants for Binomial Distribution
 24.16 Poisson Distribution
 24.17 Constants of the PoissonDistribution
 24.18 Normal Distribution
 24.19 Characteristics of the Normal Distribution
 24.20 Normal Probability Integral
 24.21 Areas Under the Standard Normal Curve
 24.22 Fitting of Normal Distribution to a Given Data
 24.23 Sampling
 24.24 Level of Significance and Critical Region
 24.25 Test of Significance for Large Samples
 24.26 Confidence Interval for the Mean
 24.27 Test of significance for Single Proportion
 24.28 Test of Significance for Difference of Proportion
 24.29 Test of Significance for Difference of Means
 24.30 Test of Significance for the Difference of Standard Deviations
 24.31 Sampling with Small Samples
 24.32 Significance Test of Difference Between Sample Means
 24.33 Chisquare Distribution
 24.34 χ2test as a Test of GoodnessofFit
 24.35 Snedecor's FDistribution
 24.36 Fisher's ZDistribution
 Exercises

25. Linear Programming
 25.1 Linear Programming Problems
 25.2 Formulation of a Linear Programming Problem (LPP)
 25.3 Graphical Method to Solve Linear Programming Problem
 25.4 Canonical and Standard forms of Linear Programming Problem
 25.5 Basic Feasible Solution of an LPP
 25.6 Simplex Method
 25.7 Tabular form of the Solution
 25.8 Generalization of Simplex Algorithm
 25.9 TwoPhase Method
 25.10 Duality Property
 25.11 Dual Simplex Method
 25.12 Transportation Problems
 25.13 Matrix form of the Transportation Problem
 25.14 Transportation Problem Table
 25.15 Basic Initial Feasible Solution of Transportation Problem
 25.16 Test for the Optimality of Basic Feasible Solution
 25.17 Degeneracy in Transportation Problem
 25.18 Unbalanced Transportation Problems
 Exercises

26. Basic Numerical Methods
 26.1 Approximate Numbers and Significant Figures
 26.2 Classical Theorems Used in Numerical Methods
 26.3 Types of Errors
 26.4 General Formula for Errors
 26.5 Solution of NonLinear Equations
 26.6 Linear System of Equations
 26.7 Finite Differences
 26.8 Error Propagation
 26.9 Interpolation
 26.10 Interpolation With Unequal Spaced Points
 26.11 Newton's Fundamental (Divided Difference) Formula
 26.12 Lagrange's Interpolation Formula
 26.13 Curve Fitting
 26.14 Numerical Quadrature (Integration)
 26.15 Ordinary Differential Equations
 26.16 Numerical Solution of Partial Differential Equations
 Exercises

12. Preliminaries
 Bibliography
 Acknowledgements
 Copyright
Product information
 Title: Engineering Mathematics
 Author(s):
 Release date: June 2009
 Publisher(s): Pearson India
 ISBN: 9788131726914
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